What I Learned Today: Scale Drawings and Maps

I asked my 15-year-old what she learned today at school. She paused for a moment and then answered my question by asking me what I learned at school today.

It took me a while to think about what I had learned [which will make me more patient when I ask her the question again tomorrow], and then I remembered and shared with her:

We are working with some teachers who are using the Illustrative Mathematics 6–8 Math curriculum. The 7^th grade teachers are in Unit 1, Scale Drawings. They are working with Scale Drawings and Maps. Today I learned to look more closely at the scale given for a map.

Look at the following for a moment. What’s the same? What’s different?

The last two are from Illustrative Mathematics, which you can download for free at openupresources.org.

What’s different about the scales on the last two?

Attend to precision, MP6, says, “Mathematically proficient students try to communicate precisely to others. … They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately.”

I’m not sure that we would have noticed a difference, except that we were trying to find some assessment items from another source and saw that many aligned to 7.G.A.1 included a scale in the form of “1 cm = 100 miles”. I’ve looked at lots of maps and I never noticed the incongruity of saying that 1 cm equals 100 miles. We don’t really mean that 1 cm equals 100 miles, right? Not in the same sense that we say 4 quarters equals $1 or 3+4=7. Is there any wonder that our students misuse the equal sign?

And so the journey continues, grateful for the authors of this curriculum who make me pay closer attention to attending to precision and grateful for my daughter who makes me think and share about what I’m learning, too …

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MP6 – Defining Terms

How do you provide your students the opportunity to attend to precision?

Writing sound definitions is a good practice for students, making all of us pay close attention to what something is and is not.

I’ve learned from Jessica Murk about Bongard Problems, which give students practice creating sound definitions based on what something is and is not.

What can you say about every figure on the left of the page that is not true about every figure on the right side of the page? (Bongard Problem #16)

Last year when I asked students to define circle, I found it hard to select and sequence the responses that would best contribute to a whole class discussion without taking too much class time.

I remember reading Dylan Wiliam’s suggestion in Embedding Formative Assessment (chapter 6, page 147) to have students give feedback to student responses that aren’t from their own class. I think it’s still helpful for students to spend time writing their own definition, and possibly trying to break a partner’s definition, but I wonder whether using some of last year’s responses to drive a whole class discussion this year might be helpful.

• a shape with no corners
• A circle is a shape that is equal distance from the center.
• a round shape whose angles add up to 360 degrees
• A circle is a two-dimensional shape, that has an infinite amount of lines of symmetry, and a total of 360 degrees.
• A 2-d figure where all the points from the center to the circumference are equidistant.

We recently discussed trapezoids.

Based on the diagram, how would you define trapezoid?

Does how you define trapezoid depend on how you construct it?

Can you construct a dynamic quadrilateral with exactly one pair of parallel sides?

And so the #AskDontTell journey continues …

 

MP6 – Mapping a Parallelogram Onto Itself

How do you provide your students the opportunity to practice I can attend to precision?

Jill and I have worked on a leveled learning progression for MP6:

Level 4:

I can distinguish between necessary and sufficient language for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

CCSS G-CO 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

We continued working on our learning intention: I can map a figure onto itself using transformations.

Perform and describe a [sequence of] transformation[s] that will map parallelogram ABCD onto itself.

This task also requires students to practice I can look for and make use of structure. What auxiliary objects will be helpful in mapping the parallelogram onto itself?

The student who shared her work drew the diagonals of the parallelogram so that she could use the intersection of the diagonals as the center of rotation.

Then she rotated the parallelogram 180˚ about that point.

Could you use only reflections to carry a parallelogram onto itself?

You can. How can you describe the sequence of reflections to carry the parallelogram onto itself?

How else could you carry a parallelogram onto itself?

 

MP6 – Mapping a Figure Onto Itself

How do you provide your students the opportunity to practice I can attend to precision?

Jill and I have worked on a leveled learning progression for MP6:

Level 4:

I can distinguish between necessary and sufficient language for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

CCSS G-CO 3:

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Our learning intention for the day was I can map a figure onto itself using transformations.

Performing a [sequence of] transformation[s] that will map rectangle ABCD onto itself is not the same thing as describing a [sequence of] transformation[s].

We practiced both, but we focused on describing.

I asked the student who listed several steps to share his work.

1. rotate rectangle 180˚ about point A
2. translate rectangle A’B’C’D’ right so that points A’ and B line up as points B’ and A. [What vector are you using?]
3. Reflect rectangle A”B”C”D” onto rectangle ABCD to get it to reflect onto itself. [About what line are you reflecting?]

What if we want to carry rectangle ABCD onto rectangle CDAB? How is this task different from just carrying rectangle ABCD onto itself?

What about mapping a regular pentagon onto itself?

Many students suggested using a single rotation, but they didn’t note the center of rotation. How could you find the center of rotation for a single rotation to map the pentagon onto itself?

This student used the intersection of the perpendicular bisectors to find the center of rotation, but didn’t know what angle to use for the rotation. How would you find an angle of rotation that would work?

What can you do other than a single rotation?

This student reflected the pentagon about the perpendicular bisectors of one of the side of the pentagon.

The descriptions students gave made it obvious that we needed more work on describing. The next day, we took some of the descriptions and critiqued them. Which students have attended to precision?

It’s good work to distinguish precision from knowing what someone means as we learn to attend to precision. And so the journey continues …

 

What’s My Rule?

We practice “I can look for and make use of structure” and “I can look for and express regularity in repeated reasoning” almost every day in geometry.

This What’s My Rule? relationship provided that opportunity, along with “I can attend to precision”.

What rule can you write or describe or draw that maps Z onto W?

As students first started looking, I heard some of the following:

• positive x axis
• x is positive, y equals 0
• they come together on (2,0)
• (?,y*0)
• when z is on top of w, z is on the positive side on the x axis

 

Students have been accustomed to drawing auxiliary objects to make use of the structure of the given objects.

As students continued looking, I saw some of the following:

Some students constructed circles with W as center, containing Z. And with Z as center, containing W.

Others constructed circles with W as center, containing the origin. And with Z as center, containing the origin.

Others constructed a circle with the midpoint of segment ZW as the center.

Another student recognized that the distance from the origin to Z was the same as the x-coordinate of W.

And then made sense of that by measuring the distance from W to the origin as well.

Does the redefining Z to be stuck on the grid help make sense of the relationship between W and Z?

 

As students looked for longer, I heard some of the following:

• The length of the line segment from the origin to Z is the x coordinate of W.
• w=((distance of z from origin),0)
• The Pythagorean Theorem

Eventually, I saw a circle with the origin as center that contained Z and W.

I saw lots of good conversation starters for our whole class discussion when I collected the student responses.

And so, as the journey continues,

Where would you start?

What questions would you ask?

How would you close the discussion?

 

0.9 Repeating

I got to teach one of my favorite lessons in a Precalculus class this week, which I developed several years ago from a paper by Thomas Osler, Fun with 0.999…

We started with a Quick Poll. Students could select as many or as few choices as they wanted.

I shared their responses separated

and grouped together.

In the first class, one student selected all three choices.

In the second class, 5 students selected all three choices.

I set the timer for a few minutes and asked students to think individually about how they could argue their selection(s).

Then I asked them to talk together about their ideas.

I walked around and listened. These are the conversations I heard:

A: 1/3 is 0.3 repeating. 2/3 is 0.6 repeating. If we add 1/3 and 2/3, we get 1. If we add 0.3 repeating and 0.6 repeating, we get 0.9 repeating.

B: 1/9 is 0.1 repeating. If we multiply 1/9 by 1, we get 1. If we multiply 0.1 repeating by 9, we get 0.9 repeating.

C: 1/3 is 0.3 repeating. If we add 1/3 three times, we get 1. If we add 0.3 repeating three times, we get 0.9 repeating.

D: If x=0.9 repeating, then 10x=9.9 repeating. (It was clear that a few students had seen Vi Hart talk about 0.9 repeating. Even so, this was all they had for now.)

E: I think this is like Zeno’s Paradox. To walk across the room, you have to walk halfway, and halfway again, and halfway again.

This was the perfect opportunity to deliberately sequence the students’ thinking and let them make connections between their arguments (5 Practices style). With which conversation would you start?

We started with argument C. More than one person shook their head in disbelief, even though they agreed that the argument was convincing.

Next we moved to argument A, which was very similar to argument C.

Next we moved to argument B.

I had a few suggestions of what to do, based on the article from the AMATYC Review. We went to one of those next that the students hadn’t thought of: If x=0.9 repeating, what happens when you divide the equation by 3?

A student shared their work differently in each class, showing that x=1.

We moved next to argument D. Again, students shared their thinking differently in each class.

No one thought about Zeno’s Paradox in the first class. So I asked them how we could express 0.9 repeating as a sum.

And then I sent a Quick Poll to collect their responses.

In the second class, I asked the students with argument E to share their thoughts. They got at the infinite sum idea, so without decomposing 0.9 repeating as a class, I sent the Quick Poll. Lots of students came up with a sum that equaled 1. Only one of those was clearly 0.9+0.99 +0.999+…

(I didn’t show them the responses equal to 1 in green when I showed them their results.)

So we practiced look for and make use of structure together. How can we decompose 0.9 repeating into a sum?

I sent the poll again.

We concluded the lesson by polling the first question again. In the first class, 4 additional students believed only that 0.9 repeating = 1 at the end.

In the second class the number of students selecting only choice A changed from 6 to 13.

Our #AskDontTell journey continues, one lesson at a time …

 

The Equation of a Circle

Expressing Geometric Properties with Equations

G-GPE.A Translate between the geometric description and the equation for a conic section

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

How do you provide an opportunity for your students to make sense of the equation of a circle in the coordinate plane? We recently use the Geometry Nspired activity Exploring the Equation of a Circle.

Students practiced look for and express regularity in repeated reasoning. What stays the same? What changes?

It’s a right triangle.

The hypotenuse is always 5.

The legs change.

What else do you notice? What has to be true for these objects?

The Pythagorean Theorem works.

How?

Leg squared plus leg squared equals five squared.

What do you notice about the legs? How can we represent the legs on the graph?

One leg is always horizontal.

One leg is always vertical.

How can we represent their lengths in the coordinate plane?

x and y?

(I think they thought that the obvious was too easy.)

What do x and y have to do with point P?

Oh! They’re the x- and y-coordinates of point P.

So what can we say is always true?

Is there an equation that is always true?

x²+y²=5²

What path does P travel? (This was preceded by – I’m going to ask a question, but I don’t want you to answer out loud. Let’s give everyone time to think.)

And then we traced point P as we moved it about coordinate plane.

So P makes a circle, and we have figured out that the equation of that circle is x²+y²=5².

I then let them explore two other pages with their teams, one where they could change the radius of the circle and one where they could change the center of the circle.

And then they answered a few questions about what they found. I used Class Capture to watch as they practiced look for and express regularity in repeated reasoning.

Here are the results of the questions that they worked.

What would you do next?

What I didn’t do at this point was differentiate my instruction. It occurred to me as soon as I got the results that I should have had a plan of what to do with the students who got 1 or 2 questions correct. It turns out that it was a team of students – already sitting together – who needed extra support – but I didn’t figure that out until later. Luckily, my students know that formative assessment isn’t just for me, the teacher – it’s for them, too. They share the responsibility in making a learning adjustment before the next class when they aren’t getting it.

We pressed on together – to make more sense out of the equation of a circle. I used a few questions from the Mathematics Assessment Project formative assessment lesson, Equations of Circles 1, getting at specific points on the circle.

And then I wondered whether we could begin making a circle. I assigned a different section of the x-y coordinate plane to each team. Send me a point (different from your team member) that lies on the circle x²+y²=64. Quadrant II is a little lacking, but overall, not too bad.

How can we graph the circle, limited to functions?

How can we tell which points are correct?

I asked them to write the equation of a circle given its center and radius, practicing attend to precision.

54% of the students were successful. The review workspace helps us attend to precision as well, since we can see how others answered.

(At the beginning of the next class, 79% of the students could write the equation, practicing attend to precision.)

I have evidence from the lesson that students are building procedural fluency from conceptual understanding (one of the NCTM Principles to Actions Mathematics Teaching Practices).

But what I liked best is that by the end of the lesson, most students reached level 4 of look for and express regularity in repeated reasoning: I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

When I asked them the equation of a circle with center (h,k) and radius r, 79% told me the standard form (or general for or center-radius form, depending on which textbook/site you use) instead of me telling them.

We closed the lesson by looking back at what happens when the circle is translated so that its center is no longer the origin. How does the right triangle change? How can that help us make sense of equation of the circle?

And so the journey continues, one #AskDontTell learning episode at a time.